Reuben David Ferguson
Senior Research Project
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Copyright 1994 Reuben David Ferguson
Twelve Scientists is a collection of twelve pieces, eleven of
which were composed using twelve-tone (serial) composition techniques.
The remaining work was composed using atonal practices, but not in a serial
manner. The pieces which constitute the collection were inspired
by the lives and works of scientists who made important contributions to
their various fields of scientific study. (The composer has had a
life-long fascination with science and, when younger, had considered entering
some area of scientific work). The number of compositions was set
at twelve to correspond with the number of pitches in an octave, and to
reflect the composition technique used. These twelve do not represent
the composer's idea of the twelve "most important" scientists; that could
only be accomplished using a much larger number of representatives, (probably
dozens), or by using extremely rigorous qualifications to reduce the number
of candidates to a much smaller number, (probably three). Rather,
these twelve represent those whose work has: (1) in some way inspired the
composer; (2) in some way lent itself to the serial technique; or (3) in
most cases, both. It must be admitted that in addition there are
one or two that were included primarily because their work was so monumental
it was felt no collection would be complete without them, (e.g. Newton).
(This does not mean that these men are uninspiring, merely that they are
All of the pieces were composed for an electronic medium, specifically synthesizer. However, an important consideration was the possibility of performance using traditional instruments at some time in the future; therefore the names on the staves use, whenever possible, the names of acoustic instruments. Traditional instrument ranges were also observed, but extreme ranges were used at times. This developed into something of a dilemma. A traditional range was desirable in light of a possible acoustic performance, but if this were strictly adhered to, it would eliminate one of the strengths of synthesis; i.e. the capability of extending acoustic sounds in a previously, (before synthesis), impossible manner. In the end, if a particular range were necessary for an effect, it was used, true acoustic range notwithstanding. (But this didn't happen very often). Using the same line of reasoning, the possibility of future acoustic performance was not allowed to prevent the composer from utilizing unusual sounds if they seemed necessary or desirable. Consequently, if some of the pieces were to be performed in an acoustic format, the ensemble would have to include a synthesizer to reproduce those sounds.
In addition, all of the pieces were composed in an electronic medium. All were realized on tape using a computer sequencer and several synthesizers; none have ever been performed "live" by human beings. Using a computer sequencer made possible a great deal of experimentation, and the composer could tell almost immediately whether or not a particular idea was appropriate. It also made possible a truly perfect performance; what is on the tape is exactly what was intended.
Some might be concerned with the degree to which the computer was involved in the actual composition of these pieces. The hardware and software played as much a part in the process of composition as Bach's quill pen did when he wrote The Brandenberg Concertos. Although there are computer programs available that can produce a rudimentary composition, this type of software is not used by the author, and played no part whatsoever in this project.
Nine synthesizers of various brands and descriptions were used in the production of Twelve Scientists, along with an IBM compatible computer and other MIDI-controlled mixers, digital delays, digital reverbs, and other necessary devices. The exact equipment used is listed separately, (see Appendix 3). Each individual work's commentary will include a list of sounds used and the synthesizers that produced them. Some sounds were "programmed" (i.e. created "from scratch") by the composer, and many were "tweaked" (i.e. adjusted for optimal effect), but the majority of sounds used are commercially available from professional programmers and/or the manufacturers of the equipment.
Each composition was written independently and is complete unto itself, as well as being a part of the entire work. Each composition uses a different ensemble of "instruments", (i.e., synthesizer programs), but many of the instruments are used in more than one piece. Hundreds of different programs were available for use; dozens of drums, basses, strings, etc. Generally, one or two of the best were chosen and used throughout the entire work in order to give it a more consistent timbre. Ninety-six different digital reverb programs were available; only three were used, (mostly only one), for the same reason. In contrast to these unifying factors, each composition is stylistically different. Heisenberg, for example, is an electric piano solo. Galileo is a fugue for pipe organ. Doppler could have been written for a (progressive) rock band; Kepler could have been written for a pops orchestra. The unity of the overall work and the differences of each constituent piece represent the large framework of science and the individuality of the scientists, each of whom was attempting to learn all he could about a small piece of the universe around us.
Werner Karl Heisenberg (1901-1976) was a German theoretical physicist.
He won the Nobel Prize in Physics in 1932 for his announcement of the uncertainty
principal and work on the structure of protons and neutrons. During
World War II, he was in charge of the German atomic bomb research program;
(just how instrumental he was in furthering or hindering this research
has recently been discussed in American newspapers and books.) Heisenberg's
work led him to form the uncertainty principal, one of the foundations
of modern quantum theory. Very simply stated, this principle says
that by measuring a subatomic particle's velocity, it becomes impossible
to ascertain the position of the particle; conversely, by determining its
position, the particle's velocity can no longer be measured. Heisenberg's
uncertainty principle was used to expand upon portions of Albert Einstein's
theories of relativity, (much to Einstein's dismay). Einstein could
not accept the idea that the physical universe is constructed upon probabilities,
not certainties, and is credited with having responded to Heisenberg with
the statement: "God does not play dice with the universe!" Einstein's
opinion notwithstanding, today the work of both Heisenberg and Einstein
form the basis of modern theoretical physics (Asimov, ii, 216-217).
Heisenberg was the first piece to be composed; in fact it was completed before the idea of doing a series occurred to the composer. It was written as a piano solo and was recorded on the accompanying tape with a Korg DW-8000 using an electric piano patch. The inspiration to begin this work was supplied by an assignment in Dr. Kenneth Keaton's Twentieth Century Music History course - i.e. write a composition using the following rules:
Since the composer had never written anything specifically atonal
before, and was uncertain how the final effort might be received, a title
pertaining to uncertainty was deemed appropriate, and Heisenberg it became.
It is the only one of the Twelve Scientists pieces that is not serial in
structure, and consequently is the only one which was composed without
using a matrix.
The rhythmic organization of the piece was "done with mirrors". The treble clef of measures 5 and 6 contain the same rhythm as do measures 1 and 2, but in retrograde, (with one small modification in measure 5). The original rhythm is reused in measures 7 and 8, and appears again, in retrograde this time, in measures 11 and 12. The bass clef is looser in construction, but the same processes were used during its development. In addition, a mirror-like rhythmic pattern is apparent between the first six measures and the last six, and each six-measure section is one three-measure section mirrored.
The pitches in the first three measures were determined using the previously listed rules. The pitches in the second three measures were obtained by transposing the previous three bars down a half step, and making necessary adjustments so as to conform to the rules. The second half of the piece, (measures 7-12), is simply the first half over again, but the melodic directions are reversed; what formerly went up, now goes down.
The new mathematical science of chaos theory, (which, among many other things, suggests that it should be theoretically possible to "unstir" green paint into its constituent blue and yellow colors), would seem to support the idea that even in systems that are apparently chaotic, order can be found.
The name Euclid of Alexandria (c. 300 BC) should be familiar to
anyone who sat through a high school geometry class. He is credited
with writing a very influential and virtually timeless book entitled Elements.
In it, Euclid stated and defined the basic terms, postulates, and axioms
of plane geometry.
A familiar aspect of any plane geometry class would be the use of pi, one of the first irrational numbers discovered. As is well known, pi is a non-repeating decimal of infinite length used to calculate the area of a circle. The value of pi has been worked out to several thousand decimal places without any sort of pattern emerging.
Like Heisenberg, Euclid was composed as the result of an assignment by Dr. Kenneth Keaton: write a twelve-tone serial composition following the rules laid down by Arnold Schoenberg. After some reflection, the author realized that a non-repeating number of infinite length seemed perfectly suited for use as a tone row, and so decided to base the composition upon pi.
Even after deciding to use pi, there was still the question of a title. There was at this time still no thought in the author's mind about composing an entire series, (that happened after Euclid was completed), but the name of a scientist was desired. Strictly speaking, any number of names could have been used; several people over many centuries have been associated with pi: Pythagoras, Eratosthenes, Archimedes, Gottfried Wilhelm von Leibniz, William Shanks, and, of course, Euclid. (Most of these names will probably be familiar to the reader except, possibly, William Shanks. In 1873, after working on the problem for fifteen years, Shanks published the value of pi to 707 places, which remained the record until 1949. In that year, the ENIAC computer calculated pi to 2035 places, and in doing so, discovered that Shanks had made an error in arithmetic, and the last hundred or so digits were wrong. In the opinion of the author, this effectively eliminated Shanks' name from consideration.) It was decided to call the composition Euclid because it was during high school classes in Euclidean geometry that the author received his first thorough indoctrination in the uses of pi (Asimov, i, 57-58).
Having decided to use pi as the basis for the composition, the author proceeded to consult literally dozens of almanacs, encyclopedias, math textbooks, and other sources of reference without being able to find a value that gave more than eight decimal places, (3.14159265). (Of course, as luck would have it, a week after completing the composition a value to forty-seven digits was found in an overlooked book in the author's own library.) As it turned out, this would not have a large effect upon the tone row.
In deciding how the digits in pi would determine the pitches of the tone row, a method similar to the one used in Pythagoras was utilized, based on intervallic relationships. However, where in Pythagoras each successive pitch was determined by its relation to the previous pitch, in Euclid each pitch would be determined based upon its distance in half-steps from the first pitch: A. For example, the first digit used was a zero. Since A is zero half-steps from A, it fit the pattern nicely. The second digit, 3, corresponded to a C, three half-steps up from A. The third (1) and fourth (4) digits yielded A# and C# respectively. The next digit of pi, a 1, was ignored because of the previous use of 1 in the series. The sixth (5), seventh (9), eighth (2), and ninth (6) digits led to D, F#, B, and D#. The next digit of pi is a five, which also had previously been used and was therefore ignored. Having run out of available digits, the last four pitches were determined by simply arranging the remaining unused four digits in numeric order: 7 8 10 11; yielding E, F, G, and G#. The complete row is thus:
0 3 1 4 5 9 2 6 7 8 10
A C A# C# D F# B D# E F G G#
Euclid begins with all four instruments simultaneously.
Synth 1 and synth 2 are each using P0; the piano and bass guitar are each
using RI3. The bass guitar part goes through a particular sequence
of rows: RI3 RI1 RI4 RI1 I5. (After measure 27, the bass alternates
between RI1 and RI4.) The piano takes the sequence even further:
RI3 P1 RI4 P1 RI5 P9 RI2 P6 RI5. Synth 1 begins by slowly moving
through P0 until measure 7. At measure 8 it somewhat belatedly picks
up a piece of pi (sic) and proceeds through R1 P4 P1 R5 and R9. The
second synthesizer part alternates between P0 and R0 until measure 15,
where the entire texture of the piece changes. The rows used at this
point are: I3 (Synth 1), I1 (Synth 2), I4 (Piano), and I5 (Bass).
Beginning at measure 26 and continuing through measure 30 is a somewhat
convoluted section which uses I3, P3, RI3 RI0, RI1, R0, and RI4 in various
voices. At this point, a return to some earlier portion of the piece
was desired, as a summation. In order to achieve the desired effect,
it was necessary that measures 32 through 36 be a repeat of measures 10
through 14. Unfortunately, that would begin the repeat in the middle
of three different rows, (R1, R0, and RI4). To circumvent this difficulty,
new material was introduced using the beginning elements of the necessary
rows, and placed so as to precede the point of repetition, but written
so as to function as a bridge between the center section and the repeat
(end). As a result, a smooth transition is achieved. The composition
is completed with a single measure in which all parts share row P0.
The time signature changes in Euclid follow the digits of pi as well. The piece begins with three measures of 3/4, followed by one measure of 1/4, four measures of 4/4, another bar of 1/4, and five measures of 5/4. At measure 15 nine measures of 9/8 begins, which coincides with the textural change mentioned earlier. Measure 24 begins two bars of 2/4, followed by six measures of 6/4. Measure 32 marks the beginning of the repeat section with five measures of 5/4. The last measure is in 3/4, which at the time was simply added to complete the song easily and logically, (the piece begins in 3/4); but as was later discovered, 3 is indeed the next digit in the sequence of pi. Thus, viewed as a whole (and adding a decimal point for clarity), the time signature changes are as follows: 3.141592653, giving the most accurate depiction of pi in the entire piece.
Pythagoras of Samos (BC 582-507 ?) was a Greek philosopher who
dealt with mathematics. Legend has it that upon discovering the theorem
that bears his name, he sacrificed a hundred oxen to the gods to show his
gratitude for the gift. Fortunately for the oxen, this probably never
took place, as he did not really discover the theorem. Pythagoras
probably supplied the first proof; the theorem was already well-known to
the Babylonians. The Pythagorean theorem states that for every right
triangle in which c is the hypotenuse, (the longest side), and a and b
are the shorter sides, then a2 + b2 = c2. The simplest example of
this relationship would be a right triangle with side lengths of 3, 4,
and 5 (3x3=9, 4x4=16, 9+16=25, which equals 5x5=25). This series,
3 4 5, is the basis for Pythagoras. (Miller, 388-389).
This theorem has found applications in a wide group of disciplines other than pure mathematics; two of interest are: art (use of the "Golden Section" is an application of the Pythagorean theorem to situate the primary figures in a painting or photograph for the most "balanced" appearance); and music (the relationship of musical pitch to the length and mass per unit length of a plucked string can be described by the theorem) (Pierce, 22-23).
Pythagoras, and his group of followers, composed a secret society that considered itself to be engaged in discovering great truths of the universe within number relationships. They deduced "triangular numbers"; studied the known regular polygons and regular solids, (and searched for new ones); and generally studied number theory. The Pythagoreans thought their interest in such arcane subjects would cause ordinary people to condemn them as sorcerers, (or worse), so their discussions and meetings were held in secret. Their discovery of a regular dodecahedron, (a 12-sided solid), that used pentagons, (another important shape for the society), in its construction was considered very privileged information, and knowledge of the existence of the dodecahedron was kept from the general public (Pierce, 23) (Asimov, i, 56).
The matrix for Pythagoras was determined a bit differently from the other matrices in that the tone row itself was not derived from a significant number; the important consideration was the intervalic distance between the notes. The dodecahedron was of primary significance to Pythagoras, so it was decided the row would begin on a D. From there, the row traveled up a major 3rd to F#, then up a perfect 4th to B, then down an augmented 5th to a D#, (of course, this is really a minor sixth, but a sixth wouldn't fit the pattern, would it?), up again a perfect 4th to G#, down a major 3rd to E, up a perfect 4th to A, and down a major 3rd to F. Up to this point, the 3 4 5 pattern had remained relatively intact; after this, a little more imagination was called for. The next two elements, G followed by C#, were considered independently in order to continue the 3 4 5 series embarked upon earlier. The intervalic distance between G and C# was considered both an augmented 4th and a diminished 5th simultaneously; this along with the addition of the final unused pitch, (A#), completed an overall pattern of:
M3 P4 aug5 P4 M3
P4 M3 - aug4th/dim5th - min7th
D F# B D# G# E A F G C# C A#
(That the final interval had to be a min7th was unavoidable at that
point, but it was deemed acceptable - seven was sometimes considered a
"magic" number by early peoples.)
Pythagoras begins with pitched bells playing row P0. The length of each note follows the 3 4 5 pattern: the first note, D, is a dotted quarter note; 3 eighth notes. The second note's (F#) total duration in eighth notes is 4; G, the third note, lasts for a total of 5 eighth notes. The pattern continues: D# = 5 eighths, G# = 4 eighths, E = 3 eighths, A = 3 eighths, F = 4 eighths, G = 5 eighths, C# = 5 eighths, C = 4 eighths, and finally, A# = 3 eighth notes in duration. This method of using the 3 4 5 pattern is repeated beginning at measure 13 in the bass part. This time, however, the basic unit is a quarter note rather than an eighth.
The order of time signature changes also reflects the 3 4 5 sequence; the piece begins with 3 measures of 3/4, followed by 4 measures of 4/4, followed by 5 measures of 5/4. The order is then reversed, yielding 5 measures of 5/4, 4 measures of 4/4, and 3 measures of 3/4. These patterns are repeated throughout the piece, with the final bar being one of 5/4.
The 3 4 5 pattern is also reflected in the choice of rows, which was restricted to rows 0, 3, 4, or 5 throughout the composition. However, these could be used in prime, retrograde, inversion, or retrograde inversion forms.
The instrumentation was chosen with some thought to the instruments that might have been available to the Pythagoreans: drums, bells, plucked strings (bass guitar and "plucks", or pizzicato strings), reed instruments (oboe and bassoon), and human voice (loosely represented by the first synthesizer).
The author's first encounter with Pythagoras and his theorem took place some years ago, at an age when anything with the name Walt Disney on it was considered absolutely fascinating, (especially if it was a cartoon!) There was an animated segment entitled: "Toot, Whistle, Plunk, and Boom", which told the story of Pythagoras and his society of mathematical musicians in a delightfully whimsical fashion. This cartoon, it must be admitted, had a great deal to do with the author's choice to compose Pythagoras.
Albert Einstein (1879-1955), the German-Swiss-American scientist
who devised the Theory of Relativity, is without a doubt the most famous
scientist in the world today. His very name has become synonymous
with genius. It is somewhat amusing to realize that this most renowned
of men is universally famous for performing work that is incomprehensible
to all but the most gifted and educated people. Yet, even those people
who haven't a clue as to the content or significance of Einstein's work
remember his name; even his first name! It was both the author's
personal admiration for this great man and the fact of the general public's
familiarity with him that led to the writing of Einstein.
The young Einstein received the Nobel Prize in Physics in 1921, not for his relativity theories, as most people assume, but for a paper on the photoelectric effect. His Special Theory of Relativity was published in 1905, after his photoelectric paper, and that in turn was followed by the General Theory of Relativity, published in 1915 (Calder, 1).
These two theories, amazing in their insight, appalling in their mathematics, can nevertheless be discussed and pondered by people of ordinary intelligence. The Special Theory of Relativity deals chiefly with gravity; the General Theory deals with high-speed motion. The central idea of the latter can be stated by the famous equation:
This equation shows that matter and energy are really two forms of the
same thing and that they can be transformed, one to the other. Mass
is really "frozen" energy! (Calder, 1-13).
Einstein was born in Ulm, Germany, the son of working-class Jewish parents. His first years of school in Munich were made difficult by financial hardship and anti-Semitism. When he was sixteen he moved to Switzerland, and there completed his doctorate. By the age of twenty-three he had a job in the Swiss patent office, (which, incidently, gave him plenty of time to think). In 1905, he wrote the paper that would win the Nobel Prize years later (Sheldon, 1-14).
Einstein felt the coming winds of war when the Nazis rose to power in Germany, and in 1930 he emigrated to the United States, where he received a hero's welcome. He received a position at Princeton University, and worked and lived there for the rest of his life (Sheldon, 1-14) (Calder, 1).
Einstein's work and its ramifications are not for the author to discuss; there are many popular books on the subject that give a very clear understanding (see Bibliography). It will suffice to say that his theories were the first real modification of the ideas of physics since Isaac Newton. Many inventions and scientific techniques owe their development to the foundation laid down by Einstein, (e.g., lasers, atomic reactors, hydrogen bombs, etc.). In addition, many predictions are made by the two theories, only some of which have been observed. The deflection of light by a "gravity lens" is one phenomenon predicted by Einstein that has been observed; the existence of "black holes" is considered almost certain, but no object yet observed can be proven to be a black hole. There are some other interesting conclusions of the two theories; the math of General Relativity says that as an object approaches the speed of light, its mass increases exponentially, until, upon reaching that absolute speed limit of the universe, its mass becomes infinite, and it would be incapable of moving at all! At the same time, the amount of energy required to accelerate the object would also increase at an exponential rate, and would also reach infinity at the speed of light. In a seeming contradiction to these examples is another that says the length of an object approaching the speed of light will progressively shorten! More commonly known are the time-dilation effects of speed-of-light travel, popularized by many science fiction books, movies, and television programs. This observed and proven effect refers to the slowing down of the passage of time during high-speed travel (Russell, 20). (Interestingly, the very agencies that serves to popularize Einstein's theories are usually guilty of ignoring or circumventing portions of them. Many young viewers of television science fiction may be shocked to find out that humans can't really travel at "warp" speeds!)
The author shares the public's veneration of this great man, both for his work in physics, and for his gentle and unassuming demeanor. He was a pacifist, (although he did have his moments of non-pacifism; Hitler comes to mind), and was reported to have been dismayed by the huge build-up of nuclear weapons by the superpowers in the late forties and early fifties. He loved to smoke his pipe, sail in his small boat, and play the piano and the violin; (he was better on the violin). He sought no publicity for himself, and was content to be left to his work (Sheldon, 10-12).
Einstein worked until his death on a "unified field theory", which would combine all of the forces of nature into one. The problem then, as now, has been gravity. Perhaps in the future, as others pursue this same problem, the legacy of his work will expand humanities' horizons beyond imagining (Calder, 1).
The tone row for Einstein was derived from the speed of light - 299,792.5 kilometers per second, or 186,282.3976 miles per second (Hoffman, 202). The first number, in km/sec, had too many nines in it for easy use as a tone row, so the author chose to use the second number. The digits to the right of the decimal would have contained too many repeated digits, so they were ignored. It must be admitted that the method of obtaining a tone row from this number seems somewhat contrived, but it worked. First, the number was written twice, one after another:
1 8 6 2 8 2 1 8 6 2 8 2
Beginning at the second 2, a one is alternately either added or subtracted:
1 8 6 2 8 2 1
8 6 2 8 2
X +1 -1 +1 -1 X -1+1
(The digits marked with an X are temporarily ignored.)
1 8 6 2 8 3 0 9
5 2 7 3
Since the third 2 was ignored and didn't get a 1 added to it,
the "neglected" 1 is also added to the last digit:
1 8 6 2 8 3 0 9
5 2 7 3
X X +1
1 8 6 2 8 3 0 9
5 2 7 4
Finally, the two ignored digits are directly replaced by the two remaining unused digits of the row, yielding the final row:
1 8 6 2 10 3 0 9 5 11 7 4
The synthesizer keyboard was then assigned numbers to obtain the actual order of pitches:
G# A# C#
A B C D E F G
0 1 2 3 4 5 6 7 8 9 10 11
(By starting the number sequence on G#, it was possible for the author
to begin the prime row with A E, for Albert Einstein, of course.)
The actual composition of Einstein was guided much more by musical considerations than by strict guidelines, as was the case in Euclid, and later in Kepler. The major musical question was: "Does it sound good?" An effort was made to manipulate the tone rows currently being used in a rhythmic fashion in order to obtain as many tonal-type chords as possible; for example, in the beginning of the piece, (section A), the first notes heard are an A minor chord, formed from elements of P0 and P3. In the next measure, an F# Major chord is formed, next a C#7(no 3rd), leading to a dotted half-note C minor in measure 5.
At this point, the String 2 part and the Bass Guitar part that doubles it begin a somewhat convoluted line in which the notes of P3 and P7 are alternated; every other note belongs to the same tone row.
Measure 9 (section B) features the entrance of the flute, heralding a sudden switch to a new texture. The flute is accompanied by tympani and pizzicato strings, and is joined briefly in measures 15-17 by an oboe for three bars of twelve-tone counterpoint. Throughout this entire section (measures 9-20), the flute is repeating a three-bar rhythmic motif with very little variation. This rhythmic motif is played a total of four times; the first two times using the pitches from row R0, and the last two times using R3.
In measures 21-43, (section C), the "pseudo-tonal" construction discussed earlier is implemented in a different way: three different tone rows are begun simultaneously in four voices - R2 (soprano), R5 (tenor), and R10 (alto and bass). The first notes of these rows form a Bb Major chord. The rows proceed in parallel fashion, locked together rhythmically as well, until at measure 24, an F Major is formed, (V in the key of Bb). The overall "harmonic progression" of measures 21 through 32 could be charted as:
23 24 25 26
27 28 29
30 31 32
|Bb | Fmi#5 |n/c |F |Ebmi#5 |Eb |Bbmi#5 A|n/c |E |G# C|D |G |
|I | v | |V |iv |IV |i * I| |V/I |III IV|V |I |
Key:Bb Bbmin A A/E G G
The key change from Bb to Bb minor (measure 27) is an abrupt modulation
to the parallel minor key, previously hinted at in measures 22 and 25.
The immediate change to the key of A is justified by use of a common tone
(C#). In measure 29, the E functions as a pivot chord from the key
of A to the key of E. In measure 30 the C which is common to both
G# (Ab) and C chords functions as a common tone for modulation to the final
key of G, which, with a IV-V-I cadence brings the 12 bar phrase to an end.
This phrase is immediately repeated in bars 33 through 44.
This "harmonic progression" is not exactly a textbook example of a proper tonal progression, of course, and is only achieved by stretching the rules rather strenuously. For example, the alert reader will recognize the Fmi(#5) as an ordinary C# chord, but then it could not have been called the v chord of the key of Bb. As flimsy a structure as it is, it does lend a certain impression of tonality, or at least consonance, to this section of Einstein.
Section C also features the entrance of the brass at measure 29; the brass then becomes the focal point until the end of the section.
On the last beat of measure 44, section D begins with the String 2 part playing a low-pitched, powerful motif based on P3, accompanied by tympani and String 1 playing pizzicato. This section, in the opinion of the composer, somehow comes close to capturing the greatness and wonder of Einstein's scientific achievement. One cannot tell in advance precisely where the line is headed, but it nonetheless has a feeling of implacability.
The piece starts a repeat section at measure 51, (A1), where the strings begin an exact repeat of the material at the start of the composition, but this time accompanied by a powerful rhythm section. At measure 59, a repeat of the D section, (D1), begins and proceeds through measure 65. The piece ends with a two-bar "tag". The overall structure of Einstein is thus:
A B C D A1 D1
It was the composer's intent while writing Einstein to not concern himself with a lot of self-imposed restrictions and numerical relationships, but instead to adhere only to the very basic rules of serial composition. In this manner, it was hoped that, perhaps, a bit more emotion could be brought out in this, the composer's tribute to his favorite scientist.
Charles Robert Darwin (1809-1882) was an English naturalist who
is remembered primarily for his book, The Origin of Species, first published
on November 24, 1859, and his theory of evolution contained therein.
His observations of the flora and fauna of the Galapagos Islands (and other
geographic areas) while serving as ship's naturalist on the H.M.S. Beagle,
formed the foundation for the theory presented in the book, more fully
titled: The Origin of Species by Means of Natural Selection, or the Preservation
of Favoured Races in the Struggle for Life. The furor that erupted
in both scientific and religious circles upon publication of this work
continues to the present day, and has had the unfortunate effect of overshadowing
some of his other work; e.g. his study of finches, and his still-definitive
work with barnacles (Darwin, 1) (Petuch, 28).
In the opinion of the author and many others, Darwin has been widely misinterpreted, misquoted, and generally misunderstood. He has been called an atheist and accused of presenting an atheistic theory; he was in fact a devout Presbyterian and delayed publication of his work for some time, correctly fearing that his theory would be misunderstood and misinterpreted as an attack upon God and the Church.
Avoiding unnecessary and inappropriate detail, it will suffice to say that after extensive thought, the composer has reconciled the seemingly opposed ideas of science and religion, at least to his own satisfaction. Consequently, the composition Darwin reflects the evolutionary ideas of science as well as acknowledging the influence of God.
The tone row for Darwin was determined, appropriately, by random chance. Dice were rolled, and the following series of numbers was obtained: 7 12 3 11 8 9 4 6 5 10 2 (eleven numbers). If the roll of the dice produced a number previously used, it was ignored and the roll repeated. Since it is impossible to roll a 0 or a 1 with a pair of dice, the necessary 1 was merely added to the end of the row. In addition, the 12 was tossed out and the 0 substituted, yielding a final row of:
7 0 3 11 8 9 4 6 5 10 2 1
In altering the results of the die rolls, the composer could be
said to have performed the same function as a "cosmic ray", causing the
"mutation" in the tone row by making these changes. However, it should
be remembered that if these changes had not been made, the result would
not have been a true tone row, and would never have been utilized by the
composer. Therefore, it can be stated that the tone row was arrived
at randomly, but in the final analysis, only certain sequences of numbers
fit the criteria for a tone row.
That particular idea seems to be a reflection of the anthropic principal, which attempts to resolve the question: "Why is the Universe as it is, rather than something else?", with the answer: "The Universe exists as we see it because if it were not
in this particular configuration, we would not exist to see it in any other configuration" (Hawking, 124).
Of course, an alternative explanation could be that the Universe exists in its present form because of the intervention, in some way, of God. Perhaps this second alternative gains credibility when one remembers that, (in the case of Darwin, at least), there was a guiding intelligence who made certain, (even though random events seemed to govern the generation of the numbers), that the number sequence would eventually become a tone row.
Darwin is scored for solo flute, two violins, a single cello, a single double bass, and a synthesized choir singing a sustained "Ahhh". (Of course, on the included tape all parts were realized using synthesizers.) It is purposely a very mournful-sounding piece, reflecting the rejection of Darwin's ideas, the loneliness of his position, and, perhaps, the uncertainty of Man's importance in the Universe.
The work begins with the fade-in of synthesized voices; (the voices represent the touch of God throughout the piece.) Immediately, a solo double bass begins, which plays Row P0 through for the first time. At the end of this row, (measure 15), the choir returns and a new offshoot of the original row begins - Row P5 - played by the second violin in parallel 4ths to the cello, which at this point begins a repeat (an octave higher) of the double bass line. These two rows proceed together, (parallel organum), until the reappearance, (in measures 27-30), of the voices and the creation of two new "species"; one a modification (mutation) of the original P0 row, the other a melody, (played by the 1st violin), which introduces new material based on row RI0 - which resembles the original P0 row only in that they are both played on stringed instruments. Soon after the entry of the voices in measures 29-30, other variations begin with the re-entry of the second violin and, in measure 33, the entry of a flute. Between measure 33 and measure 50, the voices return twice more and the various lines become more complex in their rhythmic interaction until the climax at measure 50, when the level of "specialization" becomes too top-heavy for "survival". With this last entrance of the voices, (measure 50), a mass extinction takes place, which leaves only the slowly-fading, primitive "life-form" represented by the double bass playing the original P0 tone row which began the piece.
Strict serial rules were followed during composition, with the exception of the synthesized voices part. This part was composed without regard to time; that is, the entrance of notes may not follow the sequence exactly. The chords formed are all six-note chords, (except for measures 40-43), and the proper two chords taken together represent a complete tone row. The "timelessness" of this part seemed appropriate, considering the influences that the voices were meant to represent.
The meter of the piece is not constant, but it does display a twelve-bar pattern which starts at the third measure. (Measures 1 and 2 are the only 4/4 measures in the entire piece.) Beginning with measure 3, there are seven bars of 7/4, followed by three bars of 3/4, followed by two bars of 10/4. This pattern is repeated without variation five times, which, (along with the first two measures of 4/4), encompasses the entire piece.
In 1842, Austrian scientist Christian Johann Doppler (1803-1853)
decided to try an experiment to see if his equations describing the shifting
of pitch of a moving sound source were correct. To that end, he obtained
the use of a few miles of straight railroad track and a locomotive, and
proceeded to propel a flatcar up and down the track at varying speeds for
two days. On the moving flatcar he had the 19th century equivalent
of a signal generator: an assemblage of trumpet players. In a stationary
position near the center of the run, he had the most accurate frequency-determining
equipment then available: musicians with perfect pitch. The musicians
on the ground recorded the change in pitch as the trumpets on the train
approached and receded, and Doppler's work turned out to be entirely correct
(Asimov, iii, 98).
Doppler's name has become familiar to the public in the last few years with the twin developments of "Doppler weather radar", which has the advantage of being able to detect the velocity of weather formations as well as their position, and a medical "Doppler ultrasound scan", which is similar to an X-ray photograph, but uses high-frequency sound pulses instead of electromagnetic radiation. (Dartford, 740)
These two examples of the fruits of Doppler's work are only the latest and most publicized applications; the "Doppler effect" has been utilized for quite some time in a variety of fields. Though most organists probably don't realize it, the characteristic sound of a Leslie organ speaker is due to slight pitch shifts caused by rotating speaker elements. On a grander scale, astronomers have been using Doppler's equations in their work for many years. In the United States in the 1920s, Edwin P. Hubble (1889-1953) discovered that the further away a star or other heavenly body is, the further its light emissions will be shifted toward the red end of the electromagnetic spectrum. This "red shift", due to the Doppler effect, is easily converted into a velocity. These discoveries, and the subsequent observation that almost everything in the sky is racing away from us at an increasing percentage of the speed of light, directly support the "Big Bang" theory of the creation of the Universe. (Dartford, 740) (Asimov, i, 200).
After having recalled the above information, the author was practically compelled to write Doppler. The realization that musicians had played such an important part in the experimental justification of Doppler's equations - equations that would later be expanded upon to describe the nature and creation of the universe itself - suggested musical possibilities that were simply too good to pass up. Doppler's experiment itself supplied the beginning, the end, and the unifying factor of the entire composition.
The tone row for Doppler was based upon the speed of sound, which is 1088 ft/sec at sea level at 32 degrees Fahrenheit. After converting this figure to several different scales, (e.g., meters/sec, furlongs/fortnight, etc.), the author decided to use three different measurement scales as the basic mathematical criteria for Doppler: 1088 ft/sec, (used in the bass guitar part), 742 mi/hr, (reflected in the time signatures), and 3,916,800 ft/hr, (which, with further manipulation, was used in determining the tone row). This last number, 3,916,800, was "processed" to produce a non-repeating sequence from 0 to 11.
The last zero was dropped, yielding:
3 9 1 6 8 0
A 1 was then added to each digit, yielding:
4 10 2 7 9 11
This second sequence was appended to the first, yielding:
3 9 1 6 8 0 4 10 2 7 9 11
The second 9 was replaced by the single remaining unused digit, 5, for a final tone row of:
3 9 1 6 8 0 4 10 2 7 5 11
The synthesizer keyboard was then numbered from 0 to 11, starting with
B as 0, and ascending to A# as 11. Since the tone row begins with
a 3, putting B as 0 enabled the author to begin the composition on a D,
The piece begins with the sound of trumpets and trombones in unison, panned to one side, and quickly increasing in volume as it pans across the sound field. As the sound reaches its loudest volume and the center pan position, the pitch drops an augmented 4th, (to a G#), following an exponential curve. Immediately upon completing the pitch change, the volume decreases following the same linear curve, (inverted this time), that was used to fade in. This, of course, represents Doppler's original musicians during his experiment. (One cannot help but wonder what the trumpeters thought of this job and the apparently deranged man who hired them. Unfortunately, I have not yet discovered any records or personal memoirs that might shed some light on this question.) This pattern repeats six times, arriving at the end of tone row P0 at measure 22, and sustaining this A# until measure 25.
At measure 26, a section begins which is devoid of the sustained horn parts, and is instead dominated by Hammond organ. The score indicates a switch from Brass 1 to Trombone at measure 26. When using synthesizers, the differences between these two instruments are problematic; the new designation is used only to indicate a change of register. In measure 28, the trombones begin a new motif based on P0, which is repeated rhythmically in measures 31 and 32. The entire row is repeated beginning at measure 41, except this time, the first phrase has been transposed up an octave and the second phrase down an octave. This section lasts until measure 53, whereupon the 1st brass enter again with P0, playing sustained notes with a gliss down about halfway through. The final three measures of the piece feature the brass fading away into the distance.
The rhythm of Doppler is driving and full of energy, reminiscent of the locomotive that powered the investigation. The electric bass guitar and traps are very much dependant upon each other's part. This dependance is an important feature in most rock music; even though Doppler is not in straight 4/4 time, it has a something of a rock "feel" to it, which is reinforced by the choice of instrumentation.
The pattern of time signature changes is very regular: the piece begins with seven measures of 7/4, followed by four measures of 4/4, followed by two measures of 2/4. As mentioned earlier, this is based on the speed of sound in miles per hour at sea level - 742 mph. This pattern repeats five times for a total of 65 measures; measure 66 is in 7/4, measure 67 is in 4/4, and measure 68, (the last measure of the composition), is in 2/4. This 7 4 2 pattern is also utilized in the bass part, where it determines the tone rows used - P7 followed by P4 followed by P2.
The third number used in Doppler is found in the organ part. The speed of sound is 1088 feet per second, and the organ is restricted to using only those rows that feature these digits, and in the proper order. For example, the organ begins the piece with row P10, followed by two consecutive uses of row P8.
It must be admitted that while composing Doppler, very little consideration was given to limiting the level of difficulty human players might have in performing the piece. Consequently, certain aspects of the parts would demand a high level of skill. For example, while the 1st Brass part should be possible for a human, the long sustained brass notes would require excellent breath control. The glissandi would also be difficult to perform convincingly, considering the length of time it took the author to find the proper exponential curve for the pitch bends. Both the drum and organ parts are fast, complicated and don't repeat very often, but they certainly could be played by accomplished performers.
The glissandi in Doppler should be examined further. As stated above, a great deal of thought and time went into attaining the most realistic representation of an actual "Doppler Effect", in respect to the proper pitch bend curves used. The MIDI (Musical Instrument Digital Interface) specification allows synthesizers to send and receive information which will effect a "pitch bend", which can be similar to a glissando or the bending of a guitar string, depending on how it is used. The sequencing software used during the writing of Twelve Scientists, Voyetra Sequencer Plus Gold, allows for fine control of this parameter. Possible values range from -8192 to +8192; the negative values give a descending pitch, the positive values force the pitch up. The exact correspondence of these values to real pitches is complicated by the fact that the synthesizer that is "reading" these messages can be set to respond to pitch bends at any one of twelve different levels. This is in order to facilitate real-time use of a joystick to control pitch bend; if the pitch bend parameter has been set to a major 2nd, then pushing the joystick to the extreme right, (MIDI controller value +8192), would bend the pitch up a major second higher that the note held on the keyboard. By setting the pitch bend parameter to an octave, the same displacement of the joystick, (still MIDI controller value +8192), would cause the synthesizer to raise its pitch an entire octave. By using the octave setting, a total pitch bend range of two octaves was available for use. A few minutes with a calculator yielded the following table:
Sequencer Plus Gold allows the user to "fill" a section of a track with data. The value of each insertion can be varied to increase or decrease in a predetermined manner, expressed in a percent of curvature. A curvature of 50%, for example, would result in a straight line from one pitch to the next - not a very realistic Doppler effect. This slope can be represented by graphing the equation -(x)=y using Cartesian co-ordinates:
The ideal solution was to perform the "fill" in two steps. The first data string was specified at a positive curvature of 70%, and was terminated in the center of what was to be the actual glissando. The second data string took up where the first left off with a specified curve of negative 70%. This complete process gives the effect of a glissando which begins to slowly descend in pitch, accelerating as it nears the halfway point, quickly descends past the center pitch, and progressively slows its rate of decent as time increases. A close approximation of this slope would be the graph of the equation -(y3)=x :
In addition to, or perhaps in spite of, the above specifications, the glissandi were written with some artistic licence; the author suspects that some of the bends in Doppler would require a very fast train indeed, although no mathematic analysis was performed by the author to confirm this.
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